Regression Equa

Selected Water-Gaging Stations North Dakota, M
Dakota, Minnesota, and South Dakota

By Tara Williams-Sether

U.S. Department of the Interior 

U.S. Geological Survey

In cooperation with the Bureau of Reclamation

Water-Resources Investigations Report 03-4291
U.S. Geological Survey, Reston, Virginia: 2004

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reproduce any copyrighted materials contained within this report.

ISBN

Library of Congress Cataloging-in-Publication Data
Water-Quality Summaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Regression Equations for Estimating Water-Quality Concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Equation Applications and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figures

1. Map showing location of selected gaging stations in the Red River of the North Basin,

North Dakota, Minnesota, and South Dakota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

2. Map showing land use and land cover and physiographic areas in the Red River of the

North Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4

3. Graphs showing linear and lowess relations between sulfate and streamflow for the

Red River of the North at Grand Forks, North Dakota, gaging station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Tables

1. Statistical summaries of selected water-quality constituents and streamflow (discharge)

for selected gaging stations in the Red River of the North Basin, North Dakota,

Minnesota, and South Dakota. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

2. Relations between selected water-quality constituents and streamflow (discharge)

for selected gaging stations in the Red River of the North Basin, North Dakota,

Minnesota, and South Dakota. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

By Tara Williams-Sether

Abstract

The Dakota Water Resources Act, passed by the U.S. 

Congress on December 15, 2000, authorized the Secretary of 

the Interior to conduct a comprehensive study of future waterquantity and quality needs of the Red River of the North Basin 

in North Dakota and possible options to meet those water needs. 

Previous Red River of the North Basin studies conducted by the 

Bureau of Reclamation used streamflow and water-quality data 

bases developed by the U.S. Geological Survey that included 

data for 1931-84. As a result of the recent congressional 

authorization and results of previous studies by the Bureau of 

Reclamation, redevelopment of the streamflow and waterquality data bases with current data through 1999 are needed in 

order to evaluate and predict the water-quantity and quality 

effects within the Red River of the North Basin. This report 

provides updated statistical summaries of selected waterquality constituents and streamflow and the regression relations 

between them.

Available data for 1931-99 were used to develop 

regression equations between 5 selected water-quality 

constituents and streamflow for 38 gaging stations in the Red 

River of the North Basin. The water-quality constituents that 

were regressed against streamflow were hardness (as CaCO3), 

sodium, chloride, sulfate, and dissolved solids. Statistical 

summaries of the selected water-quality constituents and 

streamflow for the gaging stations used in the regression 

equations development and the applications and limitations of 

the regression equations are presented in this report.

Introduction

In 1994, the Bureau of Reclamation (BOR) began a twophase planning study to investigate and evaluate existing and 

future municipal, rural, and industrial (MR&I) water use in the 

Red River of the North Basin in North Dakota. Phase I of the 

study assessed water needs and Phase II identified 

alternatives to address water-quantity and quality ne

I, Part A, was completed in 1998 and appraised the e

projected MR&I water needs (Bureau of Reclamati

Phase I, Part B was completed in 1999 and address

instream flow issues that pertain to aquatic life and 

quality (Bureau of Reclamation, 1999). Parts A an

Phase I study used streamflow and water-quality da

developed by the U.S. Geological Survey (USGS) th

data for 1931-84 (Guenthner and others, 1990; Gue

1991). The BOR needed the data bases to be redev

data through 1999 in order to evaluate and predict w

quantity and quality effects within the Red River o

Basin.

The Dakota Water Resources Act was passed 

Congress on December 15, 2000. The Act authoriz

Secretary of the Interior to conduct a comprehensiv

future water-quantity and quality needs of the Red R

North Basin in North Dakota and possible options to 

water needs. The BOR will administrate the comp

study. The BOR has formed a Technical Team and 

Review Team. The Technical Team will develop p

study, perform technical evaluations, draft portions 

Needs and Options Report, and perform other day-t

activities. The Study Review Team will review dra

and provide input into study processes and product

The USGS conducted a study in cooperation w

to provide needed information for the comprehensi

The study will provide updated historic and unregu

monthly streamflow for selected gaging stations in 

River of the North Basin for 1931-99 and will deve

regression equations between selected water-qualit

constituents and streamflow. Thus, the study will u

previous reports of Guenthner and others (1990) and 

(1991). The report by Emerson and Dressler (2002) 

streamflow report by Guenthner and others (1990). 

updates the water-quality and streamflow relations 

Guenthner (1991).
The equations were developed by regressing 5 selected 

water-quality constituents against streamflow for 38 selected 

gaging stations in the Red River of the North Basin (fig. 1). The 

water-quality constituents were hardness (as CaCO3), sodium, 

chloride, sulfate, and dissolved solids. Calcium, magnesium, 

nitrate, total organic carbon, iron, manganese, pH, field 

conductivity, alkalinity, and turbidity also were considered for 

the regression equations. However, data were too few or were 

considered unusable because changes in sampling techniques 

and laboratory analyses made comparison over time unrealistic. 

Data bases from the Environmental Protection Agency, North 

Dakota Department of Health, North Dakota State Water 

Commission, and various agencies from Minnesota and Canada 

(Macek-Rowland and Dressler, 2002, p. 6-7) were considered 

for the regression equations but were not used because of 

changes in sampling techniques and laboratory analyses and 

because of the absence of streamflow data. Available waterquality and streamflow data for 1931-99 were retrieved from 

the USGS computerized National Water Information System 

and from North Dakota, Minnesota, and South Dakota USGS 

data bases. Period of records for available water-quality and 

streamflow data varied within the Red River of the North Basin.

Description of Study Area

The Red River of the North Basin is part of the Hudson 

Bay drainage system. Parts of Montana, South Dakota, North 

Dakota, and Minnesota in the United States, and parts of 

Saskatchewan and Manitoba in Canada are drained by the Red 

River of the North. The North Dakota-Minnesota boundary is 

formed by the Red River of the North. Drainage area of the Red 

River of the North at Emerson, Manitoba, gaging station, which 

is 0.8 mile downstream from the international boundary, is 

40,200 square miles (U.S. Geological Survey, 2000, p. 196). 

The river flows northward 394 miles to the United StatesCanadian boundary. From the international boundary, the Red 

River of the North flows north about 155 miles and discharges 

into Lake Winnipeg. The Red River of the North Basin 

upstream from the international boundary is the only part of the 

basin included in the study area.

The Red River of the North begins at the confluence of the 

Otter Tail and Bois de Sioux Rivers in Wahpeton, N. Dak., and 

Breckenridge, Minn. The Otter Tail River is regulated by 

Dakota. The slope of the river is extremely flat. The r

only about 200 feet in its 394-mile course from Wahp

N. Dak., to the international boundary (Miller and Frink

Water quality in the river is affected by the lacustrine d

by inflow from major tributaries from North Dakota a

Minnesota, by ground-water discharge, and by munici

agricultural, and industrial effluents.

The Sheyenne River is one of the major tributarie

Red River of the North. The Sheyenne River has a dr

area of about 6,910 square miles (not including the clo

Devils Lake Basin). From its headwaters near Harvey, 

the Sheyenne River, which is about 500 miles long, fl

eastward about 150 miles, southward about 200 miles, 

northeastward to its confluence with the Red River of th

north of Fargo, N. Dak. (Souris-Red-Rainy River Basi

Commission, 1972, p. D-50).

The Sheyenne River Basin lies in two distinct 

physiographic areas (fig. 2). The drift prairie area exte

the headwaters to the vicinity of Lisbon, N. Dak., and 

glacial Lake Agassiz area extends from the vicinity of 

N. Dak., to the confluence of the Sheyenne River and 

River of the North. A hilly delta topography exists from 

City, N. Dak., to the vicinity of Kindred, N. Dak. Mo

Sheyenne River valley from the headwaters to Kindre

N. Dak., is incised into glacial till. The valley from Sh

N. Dak., to Kindred, N. Dak., ranges from 100 to 200 

depth and 0.2 to 2 miles in width. Average gradient of 

is 1.5 feet per mile in the drift prairie and hilly delta ar

about 1 foot per mile in the glacial Lake Agassiz area.

No flow has been recorded at times in the upper re

the Sheyenne River. Flow in the lower reaches of the 

regulated partly by releases from Baldhill Dam, which 

operation in 1949. Lake Ashtabula, formed by Baldhi

has a capacity of 69,100 acre-feet between the invert o

outlet conduit and the normal pool elevation. Lake A

is operated for flood control (capacity at maximum po

elevation of 1,273.2 feet is 116,500 acre-feet), municip

supply, recreation, and stream-pollution abatement thr

low-flow augmentation.
RED 

RIVER

THE

NORTH

OF 

Grand

Forks

FARGO

Wahpeton

Breckenridge

Moorhead

Emerson

Harvey

Warwick

Cooperstown

Valley City

Lisbon

Kindred

WestFargo

Shelly

Halstad

Drayton

FergusFalls

White Rock

Abercrombie

Dazey

Mapleton

Amenia

Dilworth

Kelso

Hendrum

Hillsboro

Climax

Crookston

Manvel

Minto

Arygle

Warren

Grafton

Neche

Akra

Detroit Lakes

Doran

Hickson

Enderlin

TwinValley

Red Lake Falls

MINNESOTA NORTH DAKOTA

MANITOBA

Streamflow-gagingstation and numberRed River of theNorth basin boundaryDevils Lake basinboundary

EXPLANATION

98 ̊

98 ̊

46 ̊

97 ̊

47 ̊

48 ̊

49 ̊

96 ̊

99 ̊ 100 ̊

46 ̊

Sand

Hill

 River

Marsh

 River

Two

Rivers

Grand

Marais

Creek

Buffalo

River

River

Tail

River

Red

River

Lake

Middle

River

Snake

River

River

Tamarac

North

Br.

South

Wild

Rice

River

Roseau

Rive

Clearwater

River

Lost

R. Hill

Br.

Burnham

Cr.

Mud

River

River

Thief

Buffalo

S. Br.

Cr.

Rabbit

River

Mustinka

River

Otter

Orwell

Reservoir

River Bois de Sioux

Turtle

River

Elm

River

Rush

River

River

River

Maple

Baldhill

Creek

Sheyenne

River

River

Wild

Rice

Sheyenne

 River

River

Goose

River

Pembina

Tongue

Park

River

River

Forest

Creek

Mowbray

Snowflake

Creek

Cypress

Creek

Hidden

Island

Coulee

Middle

North

Br.

Cr.

Br.

Cart

Br.

South

Little South

Beaver

Cr.

Antelope

Lake

Ashtabula

Cr.

Baldhill Dam

White Rock Dam

ReservationDam

OrwellDam

05046000

05046000

05051500

05051522

05054000

05054020

05062000

05053000

05050000

05054500

05056000

05057000

05057200

05058000

05058500

05058700

05059000

05059500

05060100

05059700

05060500

05062200 05062500

05064500

05069000

05078500

05082500

05083000

05090000

05085000

05092000

05101000

05100000

05102500

05085500

05087500

05079000

05066500

05064000

CANADA

UNITED STATES

DEVILS LAKE BASIN

NOT USED IN STUDY

Base from U.S. Geological Survey1:2,000,000, 1972

MudLake
CANADA

UNITED STATES

MINNESOTA NORTH DAKOTA

SOUTH DAKOTA

MANITOBA

 EXPLANATIONLand use and land cover Mostly cropland Cropland and grazing Cropland with woodland Woodland with cropland Forest and woodland Grassland and grazing Wetland Water Urban

98 ̊

98 ̊

46 ̊

97 ̊

47 ̊

48 ̊

97 ̊ 96 ̊ 99 ̊ 100 ̊

96 ̊

46 ̊

49 ̊

99 ̊

100 ̊

Base from U.S. Geological Survey1:2,000,000, 1972

Physiographic areas Drift Prairie Red River Valley Lake Plain Lake-Washed Till Plain MorainePhysiographic boundaryRed River of the North basin boundaryDevils Lake basin boundary

ABCD

A

B

C

D

RED 

RIVER

OF

THE

NORTH

FARGO Moorhead

East

Grand

Forks Grand

Forks

Creek

Elm

River

Rush

River

River

River

Maple

Baldhill

River

Wild

Rice

Sheyenne

 River

Goose

Antelope Cr.

Lake

Ashtabula

Turtle

River

Sheyenne

River

River

River

Pembina

Tongue

Park

River

River

Forest

Creek

Mowbray

Snowflake

Creek

Cypress

Creek

Hidden

Island

Coulee Rock

Lake

 Sand

Hill

 River

Marsh

 River

Buffalo

River

River

Tail

River

Wild

Rice

Buffalo

S. Br.

Cr.

Rabbit

River

Mustinka

River

Otter

Sioux R. Bois de

Two

Rivers

Grand

Marais

River

Red

River

Lake

Middle

River

Snake

River

River

Tamarac

North

Br.

South

River

Roseau

Riv

Clearwater

River

Lost

R.

Hill

Br.

Burnham

Cr.

Mud

River

River

Thief

MudLake

LakeTraverse

DEVILS LAKE BASIN

NOT USED IN STUDY
were less than or equal to the value given. The 75th percentile 

values may be used as indicators of concentration during lowflow conditions, and the 25th percentile values may be used as 

indicators of concentrations during high-flow conditions. 

Periods of available data for the water-quality constituents for 

each gaging station also are listed in table 1. The available 

water-quality data for each gaging station varies considerably 

throughout the Red River of the North Basin. Of the 38 gaging 

stations, about 5 percent had data available during 1940 through 

1999, about 16 percent had data available during 1950 through 

1999, about 37 percent had data available during 1960 through 

1999, about 37 percent had data available during 1970 through 

1999, about 2.5 percent had data available during 1980 through 

1999, and about 2.5 percent had data available during 1990 

through 1999. The sampling frequency of the water-quality 

constituents varied considerably at 11 of the 38 gaging stations 

but was more consistent at the remaining 27 gaging stations.

Instantaneous streamflow was the type of gaged 

streamflow desired for the regression equations because 

instantaneous streamflow provides a direct link to the waterquality constituents sampled at the same time. However, the 

records of instantaneous streamflow in the water-quality data 

base were not always complete. Daily mean streamflow 

records were more complete than instantaneous streamflow 

records and were readily available for most of the gaging 

stations; therefore, the daily mean streamflow records were 

used to develop the regression equations with the water-quality 

constituents when instantaneous streamflow records were too 

few. At some gaging stations, instantaneous and daily mean 

streamflow records were combined to complete the discharge 

record needed when only a few instantaneous streamflow 

records were incomplete.

Daily mean streamflow provides a streamflow 

measurement averaged over a day interval. Instantaneous 

streamflow provides a streamflow measurement at a point in the 

day. Daily mean streamflow may provide bias in regression 

equation development. The bias may be high or low depending 

upon the type of sample and when the sample was collected. 

For example, a high-flow sample will be collected near a peak 

flow for the day. Using a daily mean instead of instantaneous 

would under estimate the flow at the time of sampling. The 

reverse would be true for low-flow samples.

least-squares regression is used to obtain the values 

lowess plots that display a smoothed line along wit

plot of associated points. The lowess plots are hel

picking up trends in the data that a linear plot may 

linear and a lowess fit plot of sulfate and streamflow 

River of the North at Grand Forks, N. Dak., (050825

station are shown in figure 3. Many of the gaging s

similar looking plots. After reviewing the plots, it w

that a simple linear regression equation between th

of concentration and the logarithm of streamflow w

adequate to develop a relation between water-quali

constituents and streamflow for most gaging station

linear fit graph in figure 3 shows the sinuous look o

and the wide spread away from the linear line. The

general equations were used to model the relation b

water-quality constituents and streamflow.

The first equation assumed that the base-10 log

water-quality constituent concentration is a piece-w

continuous function of the base-10 logarithm of str

The breakpoints (the values of base-10 logarithm of 

at which slope changes occur) can be selected to obt

statistical fit to the data. For example, the lowess fi

indicates that a piece-wise linear relation between l

transformed sulfate concentration and log-transform

streamflow may be a reasonable assumption for the 

station. It appears that two breakpoints, one near b

logarithm of streamflow equals three and one near 

logarithm of streamflow equals four, might be appr

the data in figure 3. For the piece-wise linear mode

breakpoints, log-transformed concentration can be 

using K + 1 linear equations. For example, for two b

log-transformed concentrations can be expressed a

logC 

a0

a1 a0 –

b1 b0 – ---------------- logQ b0 ( ) – if b0 Q 

+ , ≤ log

a1

a2 a1 –

b2 b1 – ---------------- logQ b1 ( ) – if b1 Q 

+ , ≤ log

a2

a3 a2 –

b3 b2 – ---------------- logQ b2 ( ) – if b2 Q 

+ , ≤ log























=
Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio

Otter Tail River below Orwell Dam near Fergus Falls, Minn., gaging station 05046000, October 12, 1960, through August 

00060 Discharge (cubic feet per second) 41 -- -- 1,220 82 698 313 00900 Hardness, total (mg/L as CaCO3) 39 3.5 11 260 190 212 16 00930 Sodium, dissolved (mg/L as Na) 39 3.5 11 11 6.5 8.6 1.00945 Sulfate, dissolved (mg/L as SO4) 41 3.7 11 32 9.3 17 5.6 00940 Chloride, dissolved (mg/L as Cl) 41 3.7 11 14 2.7 8.3 2.9 70301 Dissolved solids, calculated, sum of constituents (mg/L) 36 3.3 11 290 214 238 18 

Bois de Sioux River near White Rock, S. Dak., gaging station 05050000, October 30, 1963, through November 2, 19

00060 Discharge (cubic feet per second) 15 -- -- 869 0.20 168 289 00900 Hardness, total (mg/L as CaCO3) 15 3.0 5 1,600 350 663 330 00930 Sodium, dissolved (mg/L as Na) 15 3.0 5 140 25 63 33 00945 Sulfate, dissolved (mg/L as SO4) 20 3.3 6 1,400 200 510 280 00940 Chloride, dissolved (mg/L as Cl) 15 3.0 5 35 6.8 17 7.7 70301 Dissolved solids, calculated, sum of constituents (mg/L) 15 3.0 5 2,360 487 977 493 

Red River of the North at Wahpeton, N. Dak., gaging station 05051500, May 2, 1973, through July 15, 1999

00060 Discharge (cubic feet per second) 53 -- -- 11,000 35 1,470 2,170 00900 Hardness, total (mg/L as CaCO3) 53 2.0 27 390 110 240 47 00930 Sodium, dissolved (mg/L as Na) 53 2.0 27 33 4.5 13 5.00945 Sulfate, dissolved (mg/L as SO4) 53 2.0 27 230 15 70 48 00940 Chloride, dissolved (mg/L as Cl) 53 2.0 27 22 1.7 10 4.3 70301 Dissolved solids, calculated, sum of constituents (mg/L) 53 2.0 27 563 154 297 70 

Red River of the North at Hickson, N. Dak., gaging station 05051522, November 3, 1975, through September 22, 19

00060 Discharge (cubic feet per second) 116 -- -- 12,000 3.0 1,010 1,830 00900 Hardness, total (mg/L as CaCO3) 116 4.6 25 800 92 270 80 00930 Sodium, dissolved (mg/L as Na) 116 4.6 25 92 6.8 16 9.7 00945 Sulfate, dissolved (mg/L as SO4) 116 4.6 25 340 5.4 80 60 
Wild Rice River near Abercrombie, N. Dak., gaging station 05053000, June 20, 1966, through September 3, 1999

00060 Discharge (cubic feet per second) 277 -- -- 9,250 0.01 217 895 00900 Hardness, total (mg/L as CaCO3) 277 8.1 34 1,300 51 510 250 00930 Sodium, dissolved (mg/L as Na) 277 8.1 34 420 5.3 120 77 00945 Sulfate, dissolved (mg/L as SO4) 240 7.1 34 1,200 11 380 220 00940 Chloride, dissolved (mg/L as Cl) 224 6.6 34 180 2.3 46 31 70301 Dissolved solids, calculated, sum of constituents (mg/L) 224 6.6 34 2,660 76 926 492 

Red River of the North at Fargo, N. Dak., gaging station 05054000, May 16, 1949, through July 13, 1999

00060 Discharge (cubic feet per second) 502 -- -- 25,800 0 1,210 2,450 00900 Hardness, total (mg/L as CaCO3) 502 11.4 44 420 85 250 50 00930 Sodium, dissolved (mg/L as Na) 499 11.6 43 43 5.2 15 5.5 00945 Sulfate, dissolved (mg/L as SO4) 438 10.0 44 270 13 73 44 00940 Chloride, dissolved (mg/L as Cl) 237 5.4 44 39 0.01 7.0 4.5 70301 Dissolved solids, calculated, sum of constituents (mg/L) 235 5.5 43 608 130 316 77 

Red River of the North below Fargo, N. Dak., gaging station 05054020, July 16, 1969, through September 16, 1986

00060 Discharge (cubic feet per second)1 180 -- -- 17,300 2.0 705 1,520 00900 Hardness, total (mg/L as CaCO3) 180 10.0 18 510 120 260 48 00930 Sodium, dissolved (mg/L as Na) 149 8.3 18 110 6.8 24 19 00945 Sulfate, dissolved (mg/L as SO4) 180 10.0 18 330 19 83 51 00940 Chloride, dissolved (mg/L as Cl) 168 9.3 18 96 4.4 15 13 70301 Dissolved solids, calculated, sum of constituents (mg/L) 134 8.9 15 741 170 357 104 

Sheyenne River above Harvey, N. Dak., gaging station 05054500, April 17, 1972, through August 25, 1999

00060 Discharge (cubic feet per second) 179 -- -- 690 0.19 22 64 00900 Hardness, total (mg/L as CaCO3) 179 6.4 28 550 58 190 86 00930 Sodium, dissolved (mg/L as Na) 179 6.4 28 480 20 240 92 00945 Sulfate, dissolved (mg/L as SO4) 179 6.4 28 560 37 220 78 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Sheyenne River near Warwick, N. Dak., gaging station 05056000, January 8, 1951, through August 5, 1999 00060 Discharge (cubic feet per second) 530 -- -- 2,310 0.10 99 272 

00900 Hardness, total (mg/L as CaCO3) 530 11.5 46 570 71 240 71 00930 Sodium, dissolved (mg/L as Na) 530 11.5 46 230 10 73 40 00945 Sulfate, dissolved (mg/L as SO4) 277 6.2 45 230 28 97 44 00940 Chloride, dissolved (mg/L as Cl) 247 5.5 45 37 0.6 12 5.7 70301 Dissolved solids, calculated, sum of constituents (mg/L) 230 5.3 43 902 136 460 160 

Sheyenne River near Cooperstown, N. Dak., gaging station 05057000, October 11, 1959, through July 20, 1999 00060 Discharge (cubic feet per second) 376 -- -- 4,150 0.02 206 510 

00900 Hardness, total (mg/L as CaCO3) 367 9.2 40 680 74 290 87 00930 Sodium, dissolved (mg/L as Na) 367 9.2 40 920 10 83 52 00945 Sulfate, dissolved (mg/L as SO4) 320 8.0 40 360 21 140 44 00940 Chloride, dissolved (mg/L as Cl) 315 7.7 41 39 0.1 16 5.7 70301 Dissolved solids, calculated, sum of constituents (mg/L) 306 7.6 40 1,360 122 552 167 

Baldhill Creek near Dazey, N. Dak., gaging station 05057200, May 2, 1972, through July 20, 1999

00060 Discharge (cubic feet per second) 96 -- -- 2,450 0.01 171 389 00900 Hardness, total (mg/L as CaCO3) 86 3.2 27 660 81 290 120 00930 Sodium, dissolved (mg/L as Na) 86 3.2 27 150 4.7 55 36 00945 Sulfate, dissolved (mg/L as SO4) 86 3.2 27 300 17 160 72 00940 Chloride, dissolved (mg/L as Cl) 96 3.4 28 51 2.1 16 11 70301 Dissolved solids, calculated, sum of constituents (mg/L) 86 3.2 27 1,020 124 492 213 

Sheyenne River below Baldhill Dam, N. Dak., gaging station 05058000, June 5, 1959, through July 14, 1999

00060 Discharge (cubic feet per second) 122 -- -- 4,710 3.6 375 824 00900 Hardness, total (mg/L as CaCO3) 112 3.3 34 380 96 230 57 00930 Sodium, dissolved (mg/L as Na) 112 3.3 34 120 20 64 19 00945 S

lf

di

l

d (

/L 

SO

) 112 3

3 34 240 48 120 39 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Sheyenne River at Valley City, N. Dak., gaging station 05058500, May 3, 1972, through April 6, 1999

00060 Discharge (cubic feet per second)1 25 -- -- 3,520 9.1 808 1,090 00900 Hardness, total (mg/L as CaCO3) 27 1.3 21 390 140 250 62 00930 Sodium, dissolved (mg/L as Na) 27 1.3 21 100 24 63 19 00945 Sulfate, dissolved (mg/L as SO4) 27 1.3 21 260 81 140 41 00940 Chloride, dissolved (mg/L as Cl) 27 1.3 21 24 5.1 15 4.7 70301 Dissolved solids, calculated, sum of constituents (mg/L) 27 1.3 21 687 244 457 118 

Sheyenne River at Lisbon, N. Dak., gaging station 05058700, August 2, 1956, through July 21, 1999

00060 Discharge (cubic feet per second) 609 -- -- 4,190 0.40 259 552 00900 Hardness, total (mg/L as CaCO3) 611 13.9 44 530 100 270 64 00930 Sodium, dissolved (mg/L as Na) 611 13.9 44 560 13 76 31 00945 Sulfate, dissolved (mg/L as SO4) 547 12.4 44 450 39 160 56 00940 Chloride, dissolved (mg/L as Cl) 333 7.6 44 110 8.0 30 15 70301 Dissolved solids, calculated, sum of constituents (mg/L) 330 7.5 44 1,040 192 531 137 

Sheyenne River near Kindred, N. Dak., gaging station 05059000, May 4, 1972, through September 2, 1999

00060 Discharge (cubic feet per second) 222 -- -- 4,070 19 366 697 00900 Hardness, total (mg/L as CaCO3) 222 7.9 28 430 120 300 61 00930 Sodium, dissolved (mg/L as Na) 222 7.9 28 110 9.5 64 18 00945 Sulfate, dissolved (mg/L as SO4) 222 7.9 28 290 50 150 39 00940 Chloride, dissolved (mg/L as Cl) 222 7.9 28 74 5.7 28 12 70301 Dissolved solids, calculated, sum of constituents (mg/L) 222 7.9 28 776 191 523 107 

Sheyenne River at West Fargo, N. Dak., gaging station 05059500, September 16, 1969, through August 31, 1999

00060 Discharge (cubic feet per second) 58 -- -- 3,050 5.9 447 688 00900 Hardness, total (mg/L as CaCO3) 58 2.1 28 400 130 280 75 00930 Sodium, dissolved (mg/L as Na) 58 2.1 28 94 23 57 17 00945 Sulfate, dissolved (mg/L as SO4) 58 2.1 28 230 8.6 130 44 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Maple River near Enderlin, N. Dak., gaging station 05059700, May 3, 1972, through July 22, 1999 00060 Discharge (cubic feet per second) 79 -- -- 3,500 1.3 226 545 

00900 Hardness, total (mg/L as CaCO3) 79 2.8 28 910 100 550 220 00930 Sodium, dissolved (mg/L as Na) 79 2.8 28 180 12 76 36 00945 Sulfate, dissolved (mg/L as SO4) 79 2.8 28 650 42 370 160 00940 Chloride, dissolved (mg/L as Cl) 79 2.8 28 140 2.5 50 30 70301 Dissolved solids, calculated, sum of constituents (mg/L) 79 2.8 28 1,540 176 881 348 

Maple River below Mapleton, N. Dak., gaging station 05060100, March 18, 1995, through July 19, 1999 00060 Discharge (cubic feet per second) 9 -- -- 6,620 0.79 1,410 2,140 

00900 Hardness, total (mg/L as CaCO3) 9 1.8 5 530 140 350 150 00930 Sodium, dissolved (mg/L as Na) 9 1.8 5 110 13 53 32 00945 Sulfate, dissolved (mg/L as SO4) 9 1.8 5 440 73 250 140 00940 Chloride, dissolved (mg/L as Cl) 9 1.8 5 59 7.4 30 18 70301 Dissolved solids, calculated, sum of constituents (mg/L) 9 1.8 5 959 204 580 281 

Rush River at Amenia, N. Dak., gaging station 05060500, May 3, 1972, through July 15, 1999

00060 Discharge (cubic feet per second) 48 -- -- 2,310 0.01 141 397 00900 Hardness, total (mg/L as CaCO3) 48 1.7 28 770 100 420 200 00930 Sodium, dissolved (mg/L as Na) 48 1.7 28 200 7 52 39 00945 Sulfate, dissolved (mg/L as SO4) 48 1.7 28 670 37 270 180 00940 Chloride, dissolved (mg/L as Cl) 48 1.7 28 120 2.3 25 21 70301 Dissolved solids, calculated, sum of constituents (mg/L) 48 1.7 28 1,390 146 655 336 

Buffalo River near Dilworth, Minn., gaging station 05062000, April 10, 1962, through March 12, 1991

00060 Discharge (cubic feet per second) 12 -- -- 5,250 6.1 596 1,520 00900 Hardness, total (mg/L as CaCO3) 11 1.4 8 600 120 380 140 00930 Sodium, dissolved (mg/L as Na) 11 1.4 8 27 4.5 17 6.6 00945 S

lf

di

l

d (

/L 

SO

) 11 1

4 8 230 36 130 60 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Elm River near Kelso, N. Dak., gaging station 05062200, February 25, 1981, through March 31, 1988

00060 Discharge (cubic feet per second)2 11 -- -- 705 0.01 182 214 00900 Hardness, total (mg/L as CaCO3) 11 1.4 8 450 120 230 120 00930 Sodium, dissolved (mg/L as Na) 11 1.4 8 59 5.9 23 18 00945 Sulfate, dissolved (mg/L as SO4) 11 1.4 8 210 35 99 67 00940 Chloride, dissolved (mg/L as Cl) 11 1.4 8 41 4.5 17 11 70301 Dissolved solids, calculated, sum of constituents (mg/L) 11 1.4 8 666 182 343 179 

Wild Rice River at Twin Valley, Minn., gaging station 05062500, September 21, 1966, through April 30, 1997

00060 Discharge (cubic feet per second) 173 -- -- 4,330 2.8 348 668 00900 Hardness, total (mg/L as CaCO3) 171 14.2 12 480 110 270 53 00930 Sodium, dissolved (mg/L as Na) 170 14.2 12 31 2.4 10 4.7 00945 Sulfate, dissolved (mg/L as SO4) 170 14.2 12 85 10 34 14 00940 Chloride, dissolved (mg/L as Cl) 170 14.2 12 50 0.6 4.5 4.2 70301 Dissolved solids, calculated, sum of constituents (mg/L) 170 14.2 12 570 147 305 64 

Wid Rice River at Hendrum, Minn., gaging station 05064000, October 1, 1962, through June 22, 1994

00060 Discharge (cubic feet per second) 118 -- -- 9,110 0.09 444 1,240 00900 Hardness, total (mg/L as CaCO3) 7 1.2 6 490 170 310 100 00930 Sodium, dissolved (mg/L as Na) 7 1.2 6 31 5.2 15 9.0 00945 Sulfate, dissolved (mg/L as SO4) 8 1.1 7 85 28 49 17 00940 Chloride, dissolved (mg/L as Cl) 118 11.8 10 31 0 5.1 3.6 70301 Dissolved solids, calculated, sum of constituents (mg/L) 7 1.2 6 593 217 366 126 

Red River of the North at Halstad, Minn., gaging station 05064500, July 8, 1961, through September 21, 1999

00060 Discharge (cubic feet per second) 163 -- -- 66,200 109 3,200 6,750 00900 Hardness, total (mg/L as CaCO3) 163 4.7 35 480 120 290 65 00930 Sodium, dissolved (mg/L as Na) 163 4.7 35 77 7.6 31 13 00945 Sulfate, dissolved (mg/L as SO4) 163 4.7 35 240 36 110 42 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Goose River at Hillsboro, N. Dak., gaging station 05066500, September 15, 1969, through August 20, 1999 00060 Discharge (cubic feet per second) 93 -- -- 5,360 0.07 409 980 

00900 Hardness, total (mg/L as CaCO3) 93 3.0 31 930 120 520 210 00930 Sodium, dissolved (mg/L as Na) 93 3.0 31 330 8.5 98 71 00945 Sulfate, dissolved (mg/L as SO4) 93 3.0 31 800 49 370 170 00940 Chloride, dissolved (mg/L as Cl) 93 3.0 31 310 5.4 70 63 70301 Dissolved solids, calculated, sum of constituents (mg/L) 93 3.0 31 2,060 184 897 412 

Sand Hill River near Climax, Minn., gaging station 05069000, November 1, 1966, through December 9, 1991 00060 Discharge (cubic feet per second) 2 -- -- 18 10 14 -- 

00900 Hardness, total (mg/L as CaCO3) 2 1.0 2 410 360 380 -- 00930 Sodium, dissolved (mg/L as Na) 2 1.0 2 19 14 16 -- 00945 Sulfate, dissolved (mg/L as SO4) 2 1.0 2 78 69 74 -- 00940 Chloride, dissolved (mg/L as Cl) 2 1.0 2 15 13 14 -- 70301 Dissolved solids, calculated, sum of constituents (mg/L) 1 1.0 1 435 -- -- -- 

Clearwater River at Red Lake Falls, Minn., gaging station 05078500, October 25, 1963, through July 18, 1995

00060 Discharge (cubic feet per second) 12 -- -- 1,300 40 452 481 00900 Hardness, total (mg/L as CaCO3) 12 2.4 5 310 120 250 56 00930 Sodium, dissolved (mg/L as Na) 12 2.4 5 11 3.4 7.9 1.9 00945 Sulfate, dissolved (mg/L as SO4) 12 2.4 5 74 16 40 19 00940 Chloride, dissolved (mg/L as Cl) 12 2.4 5 7.5 2.8 4.5 1.3 70301 Dissolved solids, calculated, sum of constituents (mg/L) 12 2.4 5 346 141 290 58 

Red Lake River at Crookston, Minn., gaging station 05079000, April 11, 1962, through April 22, 1997

00060 Discharge (cubic feet per second) 168 -- -- 15,000 70 1,420 1,900 00900 Hardness, total (mg/L as CaCO3) 167 5.4 31 390 100 210 43 00930 Sodium, dissolved (mg/L as Na) 168 5.4 31 15 2.3 5.8 2.4 00945 S

lf

di

l

d (

/L 

SO

) 168 5

4 31 120 7

0 37 24 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Red River of the North at Grand Forks, N. Dak., gaging station 05082500, June 22, 1949, through August 23, 1999

00060 Discharge (cubic feet per second) 532 -- -- 105,000 156 5,620 9,900 00900 Hardness, total (mg/L as CaCO3) 532 11.8 45 830 120 250 67 00930 Sodium, dissolved (mg/L as Na) 531 11.8 45 43 2.9 18 6.4 00945 Sulfate, dissolved (mg/L as SO4) 464 10.3 45 200 18 74 30 00940 Chloride, dissolved (mg/L as Cl) 251 5.6 45 34 0.1 10 5.0 70301 Dissolved solids, calculated, sum of constituents (mg/L) 248 5.5 45 1,890 166 327 121 

Turtle River at Manvel, N. Dak., gaging station 05083000, October 15, 1971, through April 13, 1990

00060 Discharge (cubic feet per second)1 39 -- -- 1,450 0.20 217 340 00900 Hardness, total (mg/L as CaCO3) 39 3.0 13 1,800 210 590 330 00930 Sodium, dissolved (mg/L as Na) 39 3.0 13 2,100 63 390 380 00945 Sulfate, dissolved (mg/L as SO4) 39 3.0 13 1,600 100 390 270 00940 Chloride, dissolved (mg/L as Cl) 39 3.0 13 3,600 89 610 650 70301 Dissolved solids, calculated, sum of constituents (mg/L) 39 3.0 13 8,120 449 1,750 1,410 

Forest River at Minto, N. Dak., gaging station 05085000, October 12, 1971, through August 16, 1999

00060 Discharge (cubic feet per second) 91 -- -- 6,140 0.10 203 725 00900 Hardness, total (mg/L as CaCO3) 91 3.1 29 850 110 330 100 00930 Sodium, dissolved (mg/L as Na) 91 3.1 29 170 6 41 19 00945 Sulfate, dissolved (mg/L as SO4) 91 3.1 29 290 36 150 50 00940 Chloride, dissolved (mg/L as Cl) 91 3.1 29 270 1.3 32 31 70301 Dissolved solids, calculated, sum of constituents (mg/L) 91 3.1 29 1,370 152 496 157 

Snake River at Warren, Minn., gaging station 05085500, September 16, 1953, through July 29, 1955

00060 Discharge (cubic feet per second) 15 -- -- 206 0.10 29 56 00900 Hardness, total (mg/L as CaCO3) 16 5.3 3 440 130 360 89 00930 Sodium, dissolved (mg/L as Na) 16 5.3 3 83 7.4 25 19 00945 Sulfate, dissolved (mg/L as SO4) 16 5.3 3 200 57 140 34 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
Middle River at Argyle, Minn., gaging station 05087500, April 12, 1954, through September 24, 1968 00060 Discharge (cubic feet per second) 23 -- -- 122 0.10 44 39 

00900 Hardness, total (mg/L as CaCO3) 23 7.7 3 470 120 290 81 00930 Sodium, dissolved (mg/L as Na) 23 7.7 3 14 4.3 7.7 2.6 00945 Sulfate, dissolved (mg/L as SO4) 23 7.7 3 89 25 64 18 00940 Chloride, dissolved (mg/L as Cl) 23 7.7 3 11 0 3.0 2.9 70301 Dissolved solids, calculated, sum of constituents (mg/L) 23 7.7 3 533 180 357 89 

Park River at Grafton, N. Dak., gaging station 05090000, September 22, 1969, through August 16, 1999 00060 Discharge (cubic feet per second) 80 -- -- 4,840 0.01 274 790 

00900 Hardness, total (mg/L as CaCO3) 80 2.7 30 610 120 320 120 00930 Sodium, dissolved (mg/L as Na) 80 2.7 30 370 12 100 69 00945 Sulfate, dissolved (mg/L as SO4) 80 2.7 30 420 53 210 89 00940 Chloride, dissolved (mg/L as Cl) 80 2.7 30 410 6.2 97 79 70301 Dissolved solids, calculated, sum of constituents (mg/L) 80 2.7 30 1,430 193 666 284 

Red River of the North at Drayton, N. Dak., gaging station 05092000, May 9, 1972, through August 17, 1999

00060 Discharge (cubic feet per second) 55 -- -- 91,000 169 12,300 17,700 00900 Hardness, total (mg/L as CaCO3) 55 2.0 28 480 120 250 69 00930 Sodium, dissolved (mg/L as Na) 55 2.0 28 130 7.3 39 28 00945 Sulfate, dissolved (mg/L as SO4) 55 2.0 28 220 35 89 36 00940 Chloride, dissolved (mg/L as Cl) 55 2.0 28 160 4.8 46 39 70301 Dissolved solids, calculated, sum of constituents (mg/L) 55 2.0 28 878 169 386 137 

Pembina River at Neche, N. Dak., gaging station 05100000, May 9, 1972, through August 12, 1999

00060 Discharge (cubic feet per second) 53 -- -- 10,300 5.6 1,050 1,910 00900 Hardness, total (mg/L as CaCO3) 53 1.9 28 570 98 280 110 00930 Sodium, dissolved (mg/L as Na) 53 1.9 28 59 19 39 12 00945 S

lf

di

l

d (

/L 

SO

) 53 1

9 28 250 56 150 48 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
1Discharge is combined instantaneous and daily mean streamflow. 2Discharge is instantaneous streamflow.

Tongue River at Akra, N. Dak., gaging station 05101000, May 11, 1972, through August 12, 1999

00060 Discharge (cubic feet per second) 77 -- -- 659 0.02 64 131 00900 Hardness, total (mg/L as CaCO3) 78 2.8 28 350 100 240 56 00930 Sodium, dissolved (mg/L as Na) 78 2.8 28 37 11 26 5.6 00945 Sulfate, dissolved (mg/L as SO4) 78 2.8 28 140 5.6 84 22 00940 Chloride, dissolved (mg/L as Cl) 78 2.8 28 40 2.2 10 4.5 70301 Dissolved solids, calculated, sum of constituents (mg/L) 78 2.8 28 505 170 339 67 

Red River of the North at Emerson, Manitoba, gaging station 05102500, July 30, 1977, through February 4, 1999

00060 Discharge (cubic feet per second) 143 -- -- 62,800 222 5,200 8,890 00900 Hardness, total (mg/L as CaCO3) 142 6.4 22 500 160 280 57 00930 Sodium, dissolved (mg/L as Na) 142 6.4 22 190 7.5 44 30 00945 Sulfate, dissolved (mg/L as SO4) 143 6.4 22 230 6.0 97 39 00940 Chloride, dissolved (mg/L as Cl) 143 6.4 22 240 9.8 50 43 70301 Dissolved solids, calculated, sum of constituents (mg/L) 142 6.4 22 1,060 197 435 131 

Table 1. Statistical summaries of selected water-quality constituents and streamflow (discharge) for selected gaging stations in the Red Rive

Minnesota, and South Dakota.—Continued

[mg/L, milligrams per liter; --, no data]

Descriptive statistics

Parameter

code

Property or constituent

Number

of

samples

Average

annual

sampling

frequency

Number

of

sample

years

Maximum Minimum Mean Standar

deviatio
2 3 3.5 4 4.5 5 2.5

BASE-10 LOGARITHM OF STREAMFLOW, IN CUBIC FEET PER SECOND

BASE-10 LOGARITHM OF SULFATE, IN MILLIGRAMS PER 

1

1.2

1.4

1.6

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Lowess fit

Figure 3. Linear and lowess relations between sulfate and streamflow for the Red River of the North at Grand Forks, North Dak

gaging station.
which the regression equation should be used to compute 

concentration.

If logQ is less than b0 or greater than b3, the regression 

equation should not be used to estimate concentration because 

the equation should not be used to extrapolate too far outside of 

the data used to fit the equation. For example, in the lowess fit 

in figure 3, b0 might be two and b3 might be five, and the 

regression equation could be used with confidence for 

streamflow ranging from 100 to 100,000 cubic feet per second.

The second equation used to model the relation between 

concentration and streamflow was a multiple linear regression 

model that incorporated month and streamflow as explanatory 

variables to allow the intercept and/or the slope to change 

depending upon the time of year. This model will be referred to 

as a linear model with seasonal coefficients and can be 

expressed as:

(2)

where

 is the intercept for month ;

 designates the month, with 

corresponding to January and to 

December; and

 is the slope for month .

Because the historical data for most gaging stations were not 

sufficient to estimate a separate intercept and slope for every 

month, the coefficients were expressed as linear combinations 

of various trigonometric functions of the month (Helsel and 

Hirsch, 1992).

The intercept and the slope were allowed to take different 

forms depending upon the constituent or gaging station being 

analyzed. If no seasonality was detected in the intercept or the 

slope, the regression coefficients in equation 2 would be:

and

where

, , , and are regression coefficients.

If seasonality was detected in the intercept and 

the regression coefficient in equation 2 would be:

and

where

, , , , , and are regression coef

If seasonality was detected in the intercept but 

slope, the regression coefficient in equation 2 woul

and

where

 are regression coefficients.

The piece-wise linear model and the linear mo

various forms for the intercept and slope were fitte

available data for 1931-99. Regression equations c

developed for the Maple River below Mapleton, N

(05060100); the Buffalo River near Dilworth, Minn

(05062000); the Elm River near Kelso, N. Dak., (0

and the Sand Hill River near Climax, Minn., (05069

stations because of too few available data. The Otte

below Orwell Dam near Fergus Falls, Minn., (0504

gaging station had eight data pairs sampled during 

that possibly were of a different data population or 

scatter plot of the data showed the eight data pairs 

separately from the rest of the data. The eight data 

removed prior to regression runs. Instantaneous str

and daily mean streamflow were combined to comp

discharge records at the Red River of the North bel

N. Dak., (05054020); the Sheyenne River at Valley 

logC = a m( ) + b m( )logQ

a m( ) m

m = 1 2,, , ... 12 m = 1

m = 12

b m( ) m

a m( ) a = o

b m( ) b = o

ao bo b1 b2

a m( ) ao a1

2πm

12 ----------     a2

2πm

12 ----------  

 

= + cos + sin

b m( ) bo b + 1

2πm

12 ----------     b2

2πm

12 ----------  

 

= cos + sin

ao a1 a2 bo b1 b2

ao a1

2πm

12 ----------     a2 pd2πm

12 ----------     cos + sin a3

2πm

6 ----------     

+ + cos + 

b m( ) b = o

ao ... a4 bo ,,,
statistic. The residual plots were evaluated to determine if the 

regression resulted in equal variance of the residuals for the full 

range of predicted values. Maximization of the coefficient of 

determination and F-statistic and minimization of the residual 

standard error of the regression and p-value were desired for the 

best equation. A step-wise approach was used for equations 1 

and 2 for each water-quality constituent concentration at each 

gaging station, and the best equation was selected based on 

residual variance and maximization and minimization of the 

various regression statistics. If no clear choice could be made, 

the simplest equation form was selected.

Final regression equations developed between waterquality constituent concentrations and streamflow for each 

gaging station are listed in table 2. The number of data pairs 

used in the equation development, p-value, coefficient of 

determination (R2), residual standard error, and range of 

standard error of actual concentration also are listed in table 2. 

The p-value is the probability that the tabulated F-statistic is 

greater than the calculated F-statistic and is desired to be as 

close to zero as possible. The F-statistic shows how well the 

regression equation explains the variance. The p-values ranged 

from zero to 0.244. The coefficient of determination is the 

percentage of the total variation in the base-10 logarithm of 

water-quality constituent concentration that is explained by the 

regression equation. Coefficient of determination ranged from 

0.03 to 0.97. The residual standard error is the estimated 

standard deviation of the residuals (the deviations of actual logtransformed concentrations from fitted log-transformed 

concentrations). The residual standard error ranged from 0.019 

to 0.310.

A small standard error indicates that the difference 

between the fitted and actual values of log-transformed 

concentration tend to be small. In particular, about 70 percent 

of the fitted values of log-transformed concentrations will be 

within plus or minus one standard error of the actual values. 

Equivalently, about 70 percent of the fitted values for 

untransformed concentrations (obtained by taking the anti-log 

of the fitted values for log-transformed concentrations) will 

satisfy the following inequality:

(3)

cases when concentration is underestimated, 

or overestimated, .

Because the range of standard error of actual conc

shows how far off the model predictive values can be 

fitted values, it is desirable to have a narrow range of 

error of actual concentration. Ranges of the standard 

actual concentration vary considerably by station and b

quality constituent (table 2).

Equation Applications and Limitations

Because the equations in table 2 were developed b

using mean daily streamflow or by using a combinatio

instantaneous and mean daily streamflow and because 

possible equation usage, water-quality load calculation

determined various ways. Equations will have very p

predictive ability and large error when using regressio

equations with very low R2 (less than 0.20), with high p

with high residual standard error values, and with a bro

of standard error of actual concentration (such as -40 to 

alternative to such poor regressions would be to use th

value (table 1) for the water-quality constituent being 

considered.

The following equation may be used to obtain mo

flow-weighted water-quality concentrations (FWC):

10–s– 1 cˆ – c

c ---------- 10s < < – 1

100 10( 

100 10s ( ) – 1

FWC

ci

qi

i = 1

n

∑

qi

i = 1

n

∑

= ----------------- x BCF
(R

)

Otter Tail River below Orwell Dam near Fergus Falls, Minn., gaging station 05046000

Range of fit: 208 < Q < 1,220 (without 1960’s data)

logCaCO3 = 2.37 - (0.015 - 0.006C1 - 0.008S1)logQ 29 3.18 x 10-4 0.52 0.019 

Na = 130Q-0.445 208 < Q < 468 27 5.05 x 10-7 0.81 0.022 

Na = 1.68Q0.262 468 < Q < 708

Na = 1,580Q-0.782 708 < Q < 871

Na = 1.68Q0.229 871 < Q < 1,020

Na = 189Q-0.452 1,020 < Q < 1,220

logSO4 = 0.887 + (0.084 - 0.024C1 + 0.038S1)logQ 32 0.001 0.42 0.108 

logCl = 1.63 - (0.227 - 0.020C1 - 0.017S1)logQ 31 1.66 x 10-8 0.76 0.043 

logDS = 2.43 - (0.021 - 0.005C1 - 0.008S1)logQ 28 0.001 0.50 0.020 

Bois de Sioux River near White Rock, S. Dak., gaging station 05050000

Range of fit: 0.20 < Q < 869

logCaCO3 = 2.95 - (0.136 + 0.041C1 + 0.061S1)logQ + 0.041C1 + 0.176S1 12 0.010 0.88 0.063 

logNa = 1.92 - 0.128 logQ + 0.034C1 + 0.034S1 + 0.030C2 + 0.068S2 14 0.004 0.85 0.101 

logSO4 = 2.82 - 0.136 logQ + 0.013 C1 + 0.090S1 + 0.022C2 + 0.081S2 12 0.060 0.77 0.115 

logCl = 1.22 - 0.064logQ + 0.126C1 - 0.083S1 + 0.081C2 + 0.046S2 14 6.90 x 10-5 0.95 0.055 

logDS = 3.15 - (0.145 + 0.017C1 + 0.071S1)logQ + 0.007C1 + 0.180S1 12 0.016 0.86 0.073 

Red River of the North at Wahpeton, N. Dak., gaging station 05051500

Range of fit: 35 < Q < 11,000

logCaCO3 = 2.68 - 0.024logQ + 0.332C1 + 0.019S1 + 0.218C2 + 0.024S2 50 0.085 0.19 0.088 

logNa = 1.41 - (0.116 - 0.015C1 - 0.028S1)logQ 51 0.065 0.14 0.157 

SO4 = 250Q-0.407 35 < Q < 251 48 8.81 x 10-6 0.44 0.232 

SO4 = 0.563Q0.696 251 < Q < 1,780

SO4 = 308Q-0.146 1,780 < Q < 11,000

logCl = 0.814 + (0.062 + 0.211C1 - 0.236S1)logQ - 0.678C1 + 0.734S1 45 0.008 0.32 0.174 

logDS = 2.56 - (0.041 - 0.006C1 - 0.019S1)logQ 49 0.008 0.23 0.071 
Red River of the North at Hickson, N. Dak., gaging station 05051522

Range of fit: 3.0 < Q < 12,000

CaCO3 = 760Q-0.218 3.0 < Q < 126 115 8.85 x 10-14 0.45 0.080 -17 

CaCO3 = 344Q-0.054 126 < Q < 708

CaCO3 = 56.8Q0.221 708 < Q < 2,510

CaCO3 = 27,910Q-0.571 2,510 < Q < 12,000

Na = 39.4Q-0.177 3.0 < Q < 631 114 2.59 x 10-6 0.23 0.141 -28 

Na = 2.78Q0.234 631 <Q < 2,510

Na = 1,090Q-0.528 2,510 < Q < 12,000

SO4 = 379Q-0.379 3.0 < Q < 501 115 7.05 x 10-8 0.28 0.286 -48 

SO4 = 0.142Q0.891 501 < Q < 2,510

SO4 = 10,050Q-0.536 2,510 < Q < 12,000

Cl = 24.1Q-0.206 3.0 < Q < 126 114 0.115 0.05 0.225 -40 

Cl = 6.83Q0.054 126 < Q < 2,510

Cl = 261Q-0.411 2,510 < Q < 12,000

DS = 6,670Q-0.931 3.0 < Q < 22.4 111 6.90 x 10-13 0.44 0.091 -19 

DS = 462Q-0.072 22.4 < Q < 501

DS = 113Q0.155 501 < Q < 2,510

DS = 12,960Q-0.451 2,510 < Q < 12,000

Wild Rice River near Abercrombie, N. Dak., gaging station 05053000

Range of fit: 0.01 < Q < 9,250

logCaCO3 = 2.76 - (0.088 + 0.041C1 + 0.125S1)logQ + 0.061C1 + 0.138S1 275 0 0.50 0.167 -32 

logNa = 2.14 - (0.134 + 0.012C1 + 0.166S1)logQ + 0.028C1 + 0.177S1 275 0 0.50 0.236 -42 

logSO4 = 2.64 - (0.102 + 0.021C1 + 0.171S1)logQ + 0.009C1 + 0.206S1 238 0 0.46 0.233 -41 

logCl = 1.73 - (0.111 - 0.039C1 + 0.082S1)logQ 222 0 0.47 0.230 -41 

logDS = 3.06 - (0.118 + 0.027C1 + 0.154S1)logQ + 0.022C1 + 0.192S1 222 0 0.60 0.164 -32 

pairs

conc

(pe

nation

(R2

)

error
Red River of the North at Fargo, N. Dak., gaging station 05054000

Range of fit: 0 < Q < 25,800 for Na and CaCO3 and 14 < Q < 25,800 for SO4, Cl, and DS

logCaCO3 = 2.55 - (0.061 + 0.052C1 + 0.092S1)logQ + 0.129C1 + 0.256S1 501 0 0.27 0.082 

Na = 10.5Q0.093 0 < Q < 56.2 498 0 0.18 0.139 

Na = 28.9Q -0.159 56.2 < Q < 251

Na = 2.74Q0.267 251 < Q < 1,580

Na = 290Q-0.365 1,580 < Q < 25,800

SO4 = 157Q-0.269 14 < Q < 200 437 0 0.36 0.202 

SO4 = 2.21Q0.536 200 < Q < 1,580

SO4 = 1,025Q-0.298 1,580 < Q < 25,800

logCl = 0.956 - 0.076logQ - 0.025C1 + 0.052S1 - 0.056C2 - 0.014S2 235 0.171 0.03 0.310 

(Alternate option: use mean value of 7.02 instead of equation.)

DS = 234Q0.082 14 < Q < 70.8 233 4.84 x 10-13 0.24 0.094 

DS = 538Q-0.114 70.8 < Q < 251

DS = 142Q0.128 251 < Q < 1,580

DS = 3,620Q-0.312 1,580 < Q < 25,800

Red River of the North below Fargo, N. Dak., gaging station 05054020

Range of fit: 2.0 < Q < 17,300

logCaCO3 = 2.55 - (0.057 + 0.054C1 + 0.031S1)logQ + 0.132C1 + 0.100S1 178 2.36 x 10-8 0.22 0.064 

logNa = 2.09 - (0.323 + 0.198C1 - 0.002S1)logQ + 0.418C1 + 0.038S1 149 0 0.64 0.135 

logSO4 = 2.14 - (0.143 + 0.303C1 - 0.099S1)logQ + 0.631C1 - 0.245S1 177 3.91 x 10-14 0.34 0.213 

logCl = 1.90 - (0.332 + 0.139C1 - 0.022S1)logQ + 0.322C1 + 0.005S1 167 0 0.59 0.150 

logDS = 2.84 - (0.125 + 0.109C1 + 0.026S1)logQ + 0.241C1 + 0.093S1 133 2.39 x 10-14 0.43 0.084 

Sheyenne River above Harvey, N. Dak., gaging station 05054500

Range of fit: 0.19 < Q < 690

logCaCO3 = 2.19 + (0.128 - 0.083C1 - 0.144S1)logQ + 0.038C1 + 0.033S1 178 2.40 x 10-9 0.24 0.165 

logNa = 2.48 - (0.184 - 0.002C1 + 0.139S1)logQ - 0.012C1 + 0.030S1 177 0 0.69 0.135 

SO4 = 198Q0.099 0.19 < Q < 7.9 177 0 0.38 0.143 

SO4 = 174Q0.163 7.9 < Q < 20

SO4 = 1,360Q-0.523 20 < Q < 690

logCl = 1.29 - (0.084 - 0.018C1 + 0.113S1)logQ + 0.007C1 + 0.040S1 178 0 0.40 0.146 

logDS = 2.99 - 0.117logQ + 0.007C1 - 0.038S1 + 0.092C2 - 0.027S2 177 0 0.58 0.106 

(R2

)
Sheyenne River near Warwick, N. Dak., gaging station 05056000

Range of fit: 0.10 < Q < 2,310 for Na and CaCO3 and 0.26 < Q < 2,310 for SO4, Cl, and DS

logCaCO3 = 2.44 - (0.043 + 0.013C1 + 0.126S1)logQ + 0.001C1 + 0.138S1 529 0 0.45 0.106 -22 

logNa = 1.69 + (0.142 + 0.041C1 - 0.233S1)logQ - 0.126C1 + 0.137S1 529 0 0.36 0.216 -39 

logSO4 = 1.89 + (0.066 - 0.022C1 - 0.155S1)logQ - 0.066C1 + 0.118S1 275 0 0.32 0.171 -33 

logCl = 1.04 + (0.068 + 0.046C1 - 0.234S1)logQ - 0.104C1 + 0.255S1 245 0 0.40 0.176 -33 

logDS = 2.67 + (0.012 - 0.005C1 - 0.172S1)logQ - 0.048C1 + 0.155S1 229 0 0.49 0.121 -24 

Sheyenne River near Cooperstown, N. Dak., gaging station 05057000

Range of fit: 0.02 < Q < 4,150

logCaCO3 = 2.60 - (0.093 + 0.100C1 + 0.130S1)logQ + 0.147C1 + 0.161S1 366 0 0.63 0.095 -20 

logNa = 2.05 - 0.106logQ - 0.017C1 - 0.050S1 + 0.095C2 + 0.023S2 365 0 0.40 0.151 -29 

logSO4 = 2.24 - (0.070 + 0.110C1 + 0.091S1)logQ + 0.137C1 + 0.081S1 319 0 0.41 0.118 -24 

logCl = 1.35 - (0.105 + 0.077C1 + 0.116S1)logQ + 0.118C1 + 0.146S1 314 0 0.60 0.114 -23 

logDS = 2.87 - (0.089 + 0.079C1 + 0.123S1)logQ + 0.089C1 + 0.144S1 305 0 0.61 0.099 -20 

Baldhill Creek near Dazey, N. Dak., gaging station 05057200

Range of fit: 0.01 < Q < 2,450

logCaCO3 = 2.57 - (0.091 + 0.070C1 + 0.140S1)logQ + 0.042C1 + 0.049S1 84 0 0.85 0.088 -18 

logNa = 1.86 - (0.206 + 0.057C1 + 0.147S1)logQ 84 0 0.85 0.157 -30 

logSO4 = 2.29 - (0.158 + 0.112C1 + 0.106S1)logQ 84 0 0.78 0.138 -27 

logCl = 1.32 - (0.171 + 0.037C1 + 0.065S1)logQ 95 0 0.68 0.179 -34 

logDS = 2.80 - (0.116 + 0.057C1 + 0.116S1)logQ + 0.024C1 + 0.020S1 84 0 0.85 0.094 -19 

pairs

conc

(pe

nation

(R2

)

error
Sheyenne River below Baldhill Dam, N. Dak., gaging station 05058000

Range of fit: 3.6 < Q < 4,710

logCaCO3 = 2.36 - 0.012logQ - 0.003C1 + 0.070S1 - 0.048C2 + 0.034S2 111 5.13 x 10-9 0.36 0.084 

logNa = 1.85 - 0.046logQ + 0.009C1 + 0.074S1 - 0.062C2 + 0.039S2 111 4.73 x 10-7 0.30 0.114 

logSO4 = 2.06 - 0.005logQ - 0.016C1 + 0.063S1 - 0.053C2 + 0.047S2 111 1.27 x 10-4 0.21 0.119 

logCl = 1.20 - 0.049logQ + 0.016C1 + 0.079S1 - 0.058C2 + 0.024S2 121 4.40 x 10-8 0.31 0.114 

logDS = 2.65 - 0.021logQ + 0.001C1 + 0.067S1 - 0.055C2 + 0.036S2 111 3.70 x 10-8 0.33 0.090 

Sheyenne River at Valley City, N. Dak., gaging station 05058500

Range of fit: 9.1 < Q < 3,520

CaCO3 = 106Q0.314 9.1 < Q < 63.1 22 2.65 x 10-5 0.77 0.057 

CaCO3 = 1,160Q-0.264 63.1 < Q < 398

CaCO3 = 53.1Q0.252 398 < Q < 1,260

CaCO3 = 148,800Q-0.860 1,260 < Q < 3,520

logNa = 2.38 - (0.440 + 0.122C1 - 0.416S1)logQ + 0.229C1 - 0.422S1 22 5.68 x 10-4 0.72 0.082 

logSO4 = 2.30 - (0.103 - 0.045C1 - 0.064S1)logQ 23 4.68 x 10-4 0.60 0.074 

logCl = 1.43 - (0.156 - 0.026C1 - 0.095S1)logQ 22 2.64 x 10-5 0.73 0.065 

DS = 208Q0.297 9.1 < Q < 63.1 22 1.37 x 10-5 0.79 0.059 

DS = 1,860Q-0.232 63.1 < Q < 398

DS = 117Q0.229 398 < Q < 1,260

DS = 576,600Q-0.961 1,260 < Q < 3,520

Sheyenne River at Lisbon, N. Dak., gaging station 05058700

Range of fit: 0.40 < Q < 4,190

logCaCO3 = 2.62 - 0.098logQ + 0.021C1 + 0.016S1 + 0.031C2 + 0.008S2 608 0 0.41 0.081 

logNa = 2.17 - 0.165logQ + 0.040C1 + 0.015S1 + 0.033C2 + 0.020S2 608 0 0.50 0.116 

logSO4 = 2.49 - (0.153 + 0.048S1)logQ - 0.020C1 + 0.099S1 544 0 0.38 0.126 

logCl = 1.94 - 0.260logQ + 0.026C1 + 0.011S1 + 0.020C2 - 0.001S2 330 0 0.62 0.134 

logDS = 2.92 - 0.106logQ + 0.027C1 + 0.013S1 + 0.025C2 + 0.025S2 327 0 0.45 0.087 

(R2

)
Sheyenne River near Kindred, N. Dak., gaging station 05059000

Range of fit: 19 < Q < 4,070

logCaCO3 = 2.67 - (0.088 - 0.008C1 + 0.062S1)logQ - 0.001C1 + 0.133S1 220 0 0.43 0.073 -15 

logNa = 1.98 - (0.074 - 0.046C1 + 0.094S1)logQ - 0.034C1 + 0.209S1 220 0 0.39 0.107 -22 

logSO4 = 2.29 - (0.044 - 0.001C1 + 0.102S1)logQ + 0.018C1 + 0.220S1 222 1.53 x 10-10 0.22 0.104 -21 

logCl = 2.01 - 0.276logQ + 0.011C1 + 0.016S1 + 0.010C2 + 0.008S2 220 0 0.61 0.121 -24 

logDS = 2.90 - (0.082 - 0.007C1 + 0.068S1)logQ + 0.010C1 + 0.148S1 221 0 0.41 0.076 -16 

Sheyenne River at West Fargo, N. Dak., gaging station 05059500

Range of fit: 5.9 < Q < 3,050

logCaCO3 = 2.49 - (0.004 - 0.125C1 + 0.155S1)logQ - 0.202C1 + 0.253S1 55 1.11 x 10-16 0.81 0.061 -13 

logNa = 1.88 - (0.034 - 0.134C1 + 0.189S1)logQ - 0.195C1 + 0.349S1 55 3.77 x 10-13 0.74 0.083 -17 

logSO4 = 2.26 - 0.059logQ + 0.096C1 - 0.027S1 + 0.004C2 + 0.073S2 55 4.19 x 10-6 0.48 0.104 -21 

logCl = 2.06 - (0.297 - 0.038C1 + 0.004S1)logQ 54 0 0.81 0.097 -20 

logDS = 2.84 - (0.055 - 0.079C1 + 0.165S1)logQ - 0.120C1 + 0.299S1 54 1.66 x 10-15 0.80 0.064 -14 

Maple River near Enderlin, N. Dak., gaging station 05059700

Range of fit: 1.3 < Q < 3,500

logCaCO3 = 2.92 - (0.185 + 0.067C1 + 0.052S1)logQ 77 0 0.85 0.098 -20 

logNa = 2.03 - (0.100 + 0.002C1 + 0.188S1)logQ - 0.052C1 + 0.141S1 78 0 0.78 0.139 -27 

logSO4 = 2.74 - (0.192 + 0.098C1 + 0.070S1)logQ 75 0 0.78 0.136 -27 

logCl = 1.90 - (0.218 + 0.049C1 + 0.044S1)logQ 77 0 0.69 0.179 -38 

logDS = 3.13 - (0.179 + 0.099C1 + 0.084S1)logQ + 0.044C1 + 0.056S1 77 0 0.86 0.093 -19 

Maple River below Mapleton, N. Dak., gaging station 05060100

Range of fit: 0.79 < Q < 6,620

Insufficient data for regression development.

pairs

conc

(pe

nation

(R2

)

error
Rush River at Amenia, N. Dak., gaging station 05060500

Range of fit: 0.01 < Q < 2,310

logCaCO3 = 2.77 - (0.140 + 0.117C1 + 0.113S1) logQ 45 0 0.87 0.096 

logNa = 1.82 - (0.237 + 0.138C1 + 0.149S1)logQ - 0.106C1 + 0.142S1 46 0 0.88 0.131 

logSO4 = 2.47 - (0.274 + 0.201C1 + 0.052S1)logQ - 0.177C1 + 0.105S1 44 2.01 x 10-14 0.85 0.138 

logCl = 1.71 - 0.212logQ + 0.398C1 - 0.008S1 + 0.333C2 - 0.241S2 46 7.19 x 10-9 0.68 0.222 

logDS = 2.97 - (0.145 + 0.121C1 + 0.124S1)logQ 45 0 0.88 0.101 

Buffalo River near Dilworth, Minn., gaging station 05062000

Range of fit: 6.1 < Q < 5,250

Insufficient data for regression development.

Elm River near Kelso, N. Dak., gaging station 05062200

Range of fit: 0.01 < Q < 705

Insufficient data for regression development.

Wild Rice River at Twin Valley, Minn., gaging station 05062500

Range of fit: 2.8 < Q < 4,330

logCaCO3 = 2.62 - (0.091 + 0.055C1 + 0.062S1)logQ + 0.156C1 + 0.139S1 168 0 0.72 0.045 

logNa = 1.53 - (0.272 + 0.066C1 + 0.038S1)logQ + 0.135C1 + 0.091S1 169 0 0.83 0.077 

SO4 = 64.9Q-0.159 2.8 < Q < 12.6 168 0 0.42 0.138 

SO4 = 115Q-0.386 12.6 < Q < 44.7

SO4 = 691Q-0.857 44.7 < Q < 70.8

SO4 = 0.266Q0.988 70.8 < Q < 100

SO4 = 9.18Q0.220 100 < Q < 794

SO4 = 29.9Q0.043 794 < Q < 4,330

logCl = 0.555 + 0.020logQ + 0.052C1 + 0.018S1 - 0.058C2 + 0.010S2 168 0.006 0.10 0.186 

(Alternate option: use mean value of 4.48 instead of equation.)

logDS = 2.70 - (0.100 + 0.056C1 + 0.043S1)logQ + 0.163C1 + 0.112S1 168 0 0.73 0.045 

(R2

)
Wild Rice River at Hendrum, Minn., gaging station 05064000

Range of fit: 0.09 < Q < 9,110 for Cl and 4.7 < Q < 2,160 for CaCO3, Na, SO4, and DS

CaCO3 = 628Q-0.169 7 7.38 x 10-5 0.97 0.030 -6.6 

Na = 48.6Q-0.305 7 8.00 x 10-5 0.96 0.055 -12 

logSO4 = 2.01 - (0.170 + 0.091C1 + 0.026S1)logQ 8 0.227 0.63 0.114 -23 

logCl = 0.913 - 0.127logQ + 0.053C1 + 0.074S1 - 0.008C2 - 0.005S2 114 1.11 x 10-15 0.52 0.128 -25 

DS = 728Q-0.165 7 1.51 x 10-4 0.95 0.034 -7.4 

Red River of the North at Halstad, Minn., gaging station 05064500

Range of fit: 109 < Q < 66,200

logCaCO3 = 2.75 - 0.098logQ + 0.032C1 + 0.014S1 + 0.047C2 - 0.019S2 163 0 0.43 0.084 -18 

logNa = 2.16 - 0.229logQ + 0.028C1 + 0.009S1 + 0.053C2 - 0.046S2 162 0 0.53 0.136 -27 

logSO4 = 2.16 - 0.047logQ + 0.027C1 - 0.028S1 + 0.083C2 - 0.064S2 161 7.76 x 10-7 0.21 0.159 -31 

logCl = 1.95 - 0.243logQ + 0.027C1 + 0.016S1 + 0.039C2 - 0.051S2 161 0 0.53 0.138 -27 

logDS = 2.94 - 0.109logQ + 0.024C1 + 0.010S1 + 0.047C2 -0.024S2 161 0 0.43 0.087 -18 

Goose River at Hillsboro, N. Dak., gaging station 05066500

Range of fit: 0.07 < Q < 5,360

logCaCO3 = 2.88 - (0.101 + 0.044C1 + 0.140S1)logQ + 0.060C1 + 0.168S1 90 0 0.80 0.100 -20 

logNa = 2.29 - (0.239 + 0.072C1 + 0.167S1)logQ + 0.134C1 + 0.262S1 89 0 0.86 0.136 -27 

logSO4 = 2.75 - (0.124 + 0.057C1 + 0.154S1)logQ + 0.075C1 + 0.194S1 90 0 0.78 0.122 -26 

logCl = 2.21 - (0.335 + 0.066C1 + 0.145S1)logQ + 0.149C1 + 0.315S1 91 0 0.86 0.164 -31 

logDS = 3.16 - (0.142 + 0.057C1 + 0.140S1)logQ + 0.088C1 + 0.192S1 89 0 0.86 0.095 -20 

Sand Hill River near Climax, Minn., gaging station 05069000

Range of fit: 10 < Q < 18 for CaCO3, Na, SO4, and Cl and 18 < Q < 18 for DS

Insufficient data for regression development.

pairs

conc

(pe

nation

(R2

)

error
Clearwater River at Red Lake Falls, Minn., gaging station 05078500

Range of fit: 40 < Q < 1,300

logCaCO3 = 2.73 - (0.158 + 0.143C1 + 0.134S1)logQ + 0.249C1 + 0.279S1 12 0.244 0.60 0.102 

logNa = 1.15 - (0.103 + 0.004C1 - 0.013S1)logQ 11 0.047 0.66 0.050 

logSO4 = 1.36 + 0.098logQ - 0.048C1 - 0.092S1 + 0.102C2 - 0.148S2 11 0.073 0.80 0.125 

logCl = 0.716 + (0.009 + 0.475C1 - 0.072S1)logQ - 1.02C1 + 0.266S1 11 0.194 0.69 0.078 

logDS = 2.49 + (0.008 + 0.018C1 + 0.038S1)logQ + 0.033C1 - 0.009S1 10 0.194 0.76 0.036 

Red Lake River at Crookston, Minn., gaging station 05079000

Range of fit: 70 < Q < 15,000

logCaCO3 = 2.62 - (0.106 + 0.079C1 + 0.044S1)logQ + 0.216C1 + 0.112S1 166 3.33 x 10-16 0.40 0.067 

logNa = 1.54 - 0.276logQ - 0.017C1 - 0.017S1 + 0.014C2 - 0.031S2 167 0 0.62 0.100 

logSO4 = 1.80 - 0.115logQ - 0.195C1 - 0.96S1 + 0.032C2 - 0.163S2 166 0 0.43 0.219 

Cl = 41.6Q-0.365 70 < Q < 631 165 3.09 x 10-13 0.33 0.166 

Cl = 40.1Q-0.359 631 < Q < 1,260

Cl = 0.147Q0.427 1,260 < Q < 3,980

Cl = 33.7Q-0.229 3,980 < Q < 15,000

logDS = 2.68 - (0.110 + 0.081C1 + 0.036S1)logQ + 0.212C1 + 0.089S1 166 2.33 x 10-15 0.38 0.072 

Red River of the North at Grand Forks, N. Dak., gaging station 05082500

Range of fit: 156 < Q < 105,000

logCaCO3 = 2.70 - 0.092logQ - 0.012C1 + 0.007S1 + 0.051C2 - 0.013S2 528 0 0.40 0.066 

logNa = 1.88 - (0.203 + 0.116C1 + 0.048S1)logQ + 0.336C1 + 0.200S1 528 0 0.35 0.126 

logSO4 = 1.86 - 0.015logQ - 0.116C1 - 0.015S1 + 0.068C2 - 0.061S2 464 0 0.30 0.153 

logCl = 1.59 - (0.185 + 0.001C1 - 0.013S1)logQ 249 2.06 x 10-12 0.21 0.178 

DS = 1,670Q-0.244 156 < Q < 1,260 246 0 0.32 0.079 

DS = 103Q0.146 1,260 < Q < 5,010

DS = 2,230Q-0.215 5,010 < Q < 105,000

(R2

)
Turtle River at Manvel, N. Dak., gaging station 05083000

Range of fit: 0.20 < Q < 1,450

logCaCO3 = 2.94 - (0.097 - 0.006C1 + 0.075S1)logQ 36 2.33 x 10-10 0.77 0.104 -21 

logNa = 2.83 - (0.163 + 0.016C1 + 0.104S1)logQ 36 6.02 x 10-9 0.72 0.172 -33 

logSO4 = 2.72 - (0.080 - 0.013C1 + 0.078S1)logQ 36 3.93 x 10-7 0.63 0.136 -27 

logCl = 2.96 - (0.120 + 0.002C1 + 0.124S1)logQ 37 4.31 x 10-8 0.67 0.194 -36 

logDS = 3.41 - (0.101 - 0.017C1 + 0.094S1)logQ 38 1.61 x 10-9 0.72 0.139 -27 

Forest River at Minto, N. Dak., gaging station 05085000

Range of fit: 0.10 < Q < 6,140

logCaCO3 = 2.63 - (0.083 + 0.058C1 + 0.097S1)logQ + 0.071C1 + 0.104S1 90 0 0.69 0.082 -17 

Na = 56.0Q-0.178 0.10 < Q < 20.0 88 4.50 x 10-9 0.39 0.126 -25 

Na = 23.2Q0.118 20.0 < Q < 158

Na = 164Q-0.269 158 < Q < 6,140

SO4 = 149Q-0.125 0.10 < Q < 4.0 91 5.70 x 10-6 0.31 0.138 -27 

SO4 = 87.6Q0.259 4.0 < Q < 9.0

SO4 = 271Q-0.257 9.0 < Q < 20.0

SO4 = 80.6Q0.148 20.0 < Q < 200

SO4 = 1,440Q-0.397 200 < Q < 6,140

Cl = 52.1Q-0.152 0.10 < Q < 4.0 89 0 0.76 0.122 -25 

Cl = 84.8Q-0.505 4.0 < Q < 20.0

Cl = 15.7Q0.057 20.0 < Q < 200

Cl = 207Q-0.430 200 < Q < 6,140

DS = 596Q-0.132 0.10 < Q < 3.0 89 4.44 x 10-16 0.58 0.086 -18 

DS = 531Q-0.033 3.0 < Q < 200

DS = 2,170Q-0.298 200 < Q < 6,140

Snake River at Warren, Minn., gaging station 05085500

Range of fit: 0.10 < Q < 206

logCaCO3 = 2.40 - (0.012 - 0.087C1 + 0.137S1)logQ - 0.299C1 + 0.072S1 13 0.001 0.92 0.036 -7.9 

logNa = 2.54 - (0.623 + 0.516C1 + 0.393S1)logQ + 1.17C1 + 0.104S1 13 0.004 0.88 0.060 -13 

logSO4 = 2.24 - (0.698 + 0.628C1 - 0.220S1)logQ 14 1.44 x 10-4 0.86 0.059 -13 

Cl = 12.7Q-0.362 0.10 < Q < 7.94 13 1.56 x 10-5 0.89 0.185 -35 

Cl = 30.6Q-0.788 7.94 < Q < 206

logDS = 2.77 - (0.146 + 0.073C1 + 0.143S1)logQ - 0.025C1 + 0.067S1 13 0.005 0.87 0.041 -9.0 

pairs

conc

(pe

nation

(R2

)

error
Middle River at Argyle, Minn., gaging station 05087500

Range of fit: 0.10 < Q < 122

logCaCO3 = 2.50 - (0.085 + 0.108C1 + 0.106S1)logQ 23 1.46 x 10-5 0.72 0.080 

Na = 12.8Q0.042 0.10 < Q < 2.0 22 0.003 0.64 0.100 

Na = 23.8Q-0.857 2.0 < Q < 7.94

Na = 1.16Q0.602 7.94 < Q < 25.1

Na = 16.0Q-0.214 25.1 < Q < 70.8

Na = 23.1Q-0.299 70.8 < Q < 122

logSO4 = 1.43 - (0.183 + 0.292C1 - 0.165S1)logQ - 0.286C1 - 0.077S1 22 0.012 0.57 0.107 

logCl = 3.01 - 0.562logQ + 3.10C1 + 0.171S1 + 1.14C2 - 0.218S2 18 3.93 x 10-5 0.88 0.124 

logDS = 2.56 - (0.071 + 0.121C1 + 0.091S1)logQ 23 1.21 x 10-5 0.73 0.069 

Park River at Grafton, N. Dak., gaging station 05090000

Range of fit: 0.01 < Q < 4,840

logCaCO3 = 2.59 - (0.045 + 0.020C1 + 0.100S1)logQ + 0.020C1 - 0.001S1 78 0 0.81 0.080 

Na = 115Q-0.066 0.01 < Q < 1.58 79 0 0.81 0.142 

Na = 121Q0.052 1.58 < Q < 3.98

Na = 177Q-0.220 3.98 < Q < 25.1

Na = 263Q-0.344 25.1 < Q < 4,840

logSO4 = 2.41 - (0.062 + 0.014C1 + 0.113S1)logQ 79 0 0.77 0.105 

Cl = 123Q-0.050 0.01 < Q < 6.31 79 0 0.78 0.199 

Cl = 184Q-0.270 6.31 < Q < 35.5

Cl = 379Q-0.472 35.5 < Q < 4,840

logDS = 2.93 - (0.074 - 0.019C1 + 0.105S1)logQ - 0.021C1 + 0.024S1 79 0 0.83 0.093 

(R2

)
Red River of the North at Drayton, N. Dak., gaging station 05092000

Range of fit: 169 < Q < 91,000

CaCO3 = 540Q-0.091 169 < Q < 1,260 55 8.80 x 10-8 0.56 0.086 -18 

CaCO3 = 943Q-0.169 1,260 < Q < 3,160

CaCO3 = 185Q0.033 3,160 < Q < 7,080

CaCO3 = 259Q-0.005 7,080 < Q < 15,850

CaCO3 = 15,210Q-0.426 15,850 < Q < 91,000

logNa = 3.13 - (0.455 - 0.035S1)logQ 53 1.34 x 10-13 0.72 0.151 -29 

SO4 = 175Q-0.074 169 < Q < 1,000 54 0.015 0.22 0.156 -30 

SO4 = 1,010Q-0.328 1,000 < Q < 3,160

SO4 = 5.45Q0.320 3,160 < Q < 7,940

SO4 = 874Q-0.245 7,940 < Q < 91,000

Cl = 487Q-0.261 169 < Q < 1,000 53 3.33 x 10-15 0.78 0.174 -33 

Cl = 305,350Q-1.19 1,000 < Q < 2,000

Cl = 137Q-0.179 2,000 < Q < 17,780

Cl = 223,070Q-0.935 17,780 < Q < 91,000

DS = 1,890Q-0.193 169 < Q < 1,120 53 8.11 x 10-11 0.66 0.092 -19 

DS = 5,540Q-0.346 1,120 < Q < 2,820

DS = 282Q0.029 2,820 < Q < 10,000

DS = 5,980Q-0.303 10,000 < Q < 91,000

Pembina River at Neche, N. Dak., gaging station 05100000

Range of fit: 5.6 < Q < 10,300

logCaCO3 = 2.68 - (0.106 + 0.013C1 + 0.044S1)logQ 52 1.11 x 10-16 0.80 0.082 -17 

logNa = 1.73 - (0.051 + 0.008C1 + 0.085S1)logQ - 0.002C1 + 0.106S1 52 2.15 x 10-11 0.71 0.079 -17 

logSO4 = 2.40 - (0.109 + 0.108C1 + 0.071S1)logQ + 0.146C1 + 0.064S1 52 1.02 x 10-9 0.66 0.096 -20 

logCl = 1.54 - (0.198 + 0.047C1 + 0.034S1)logQ 52 1.33 x 10-15 0.78 0.107 -22 

logDS = 2.85 - (0.093 + 0.021C1 + 0.041S1)logQ 52 0 0.82 0.066 -14 

pairs

conc

(pe

nation

(R2

)

error
Tongue River at Akra, N. Dak., gaging station 05101000

Range of fit: 0.02 < Q < 659

logCaCO3 = 2.44 - (0.052 + 0.028C1 + 0.068S1)logQ + 0.063C1 + 0.066S1 75 3.84 x 10-11 0.56 0.071 

logNa = 1.44 - (0.017 + 0.025C1 + 0.053S1)logQ + 0.038C1 + 0.006S1 76 1.97 x 10-6 0.38 0.079 

logSO4 = 1.96 - (0.037 + 0.069C1 + 0.051S1)logQ + 0.039C1 + 0.012S1 75 0.001 0.26 0.098 

logCl = 0.960 + (0.040 + 0.031C1 - 0.076S1)logQ - 0.051C1 + 0.041S1 75 0.008 0.20 0.107 

logDS = 2.59 - (0.044 + 0.038C1 + 0.062S1)logQ + 0.052C1 + 0.054S1 74 7.43 x 10-10 0.52 0.065 

Red River of the North at Emerson, Manitoba, gaging station 05102500

Range of fit: 222 < Q < 62,800

logCaCO3 = 2.72 - (0.087 + 0.061C1 + 0.053S1)logQ + 0.208C1 + 0.181S1 142 1.62 x 10-11 0.35 0.073 

Na = 5,130Q-0.685 222 < Q < 1,780 140 0 0.58 0.156 

Na = 10.3Q0.145 1,780 < Q < 3,980

Na = 357Q-0.282 3,980 < Q < 62,800

logSO4 = 2.23 - 0.081logQ - 0.031C1 - 0.026S1 + 0.096C2 - 0.077S2 141 2.35 x 10-9 0.30 0.144 

Cl = 17,210Q-0.847 222 < Q < 1,780 142 0 0.59 0.198 

Cl = 7.52Q0.187 1,780 < Q < 3,550

Cl = 723Q-0.372 3,550 < Q < 62,800

DS = 3,820Q-0.311 222 < Q < 2,000 141 0 0.48 0.085 

DS = 46.9Q0.268 2,000 < Q < 3,980

DS = 1,670Q-0.163 3,980 < Q < 62,800

(R2

)
, where s is the residual standard error.

After the FWC is determined, a load may be calculated using 

the following equation:

Load = FWC x cf x (5)

where

Load is monthly constituent load, dependent upon units 

desired (for example, tons per acre-foot);

cf is conversion factor to change FWC into desired units 

(for example, cf = 0.00136 to change FWC from 

milligrams per liter to tons per acre-foot); and

 is total streamflow for month, dependent upon units 

desired (for example, acre-foot).

An estimate of FWC that occurs at the mean flow for a 

given month may be determined using equation 4. Instead of 

using mean daily streamflow values, a single monthly mean 

streamflow value is used for qi. This method is not accurate for 

proper load estimates, but may be adequate if a point-value for 

a given month is desired. However, additional monthly and/or 

seasonal bias would need to be determined to more accurately 

estimate the load because this method could over or under 

estimate the actual monthly flow-weighted water-quality 

concentration.

Summary

The Dakota Water Resources Act was passed by the U.S. 

Congress on December 15, 2000. The Act authorized the 

Secretary of the Interior to conduct a comprehensive study, 

administrated by the Bureau of Reclamation, of future waterquantity and quality needs of the Red River of the North Basin 

in North Dakota and possible options to meet those water needs. 

To provide needed information for the comprehensive study, 

the U.S. Geological Survey conducted a study in cooperation 

with the Bureau of Reclamation. This report provides updated 

statistical summaries of selected water-quality constituents and 

streamflow and the regression relations between them.

Available data for 1931-99 were used to develop 

regression equations between 5 selected water-quality 

and daily mean streamflow records were combined to c

the streamflow record needed. Statistical summaries o

selected water-quality constituents and streamflow for 

selected gaging stations used in the regression equatio

development are presented in this report. The statistic

summaries include number of samples; average annua

sampling frequency; number of samples years; descrip

statistics of the maximum, minimum, mean, and stand

deviation; and values for the 25th, 50th, and 75th perc

Two general equations were considered to model 

relation between water-quality constituents and stream

the selected gaging stations in the Red River of the Nort

The first equation assumed that the base-10 logarithm 

water-quality constituent concentration is a piece-wise 

continuous function of the base-10 logarithm of stream

The second equation was a multiple linear regression m

incorporated month and streamflow as explanatory var

allow the intercept and/or slope to change depending u

time of year. The best equation for each gaging statio

selected based on review of the residual plots, coeffici

determination, residual standard error of the regressio

value, and F-statistic. The final regression equations, 

of data pairs used in the equation development, p-valu

coefficient of determination, residual standard error, a

of standard error of actual concentration for each gagin

are presented in this report. The applications and limit

the regression equations also are presented in this repo

References

Bureau of Reclamation, 1998, Red River Valley Wate

Assessment--Phase I Part A, MR&I Appraisal Repo

Department of the Interior, 138 p.

Bureau of Reclamation, 1999, Red River Valley Wate

Assessment--Final Appraisal Report, Phase I Part B

Instream Flow Needs Assessment: U.S. Departmen

Interior, 83 p.

Emerson, D.G. and Dressler, V.M., 2002, Historic and 

unregulated monthly streamflow for selected sites in 

River of the North Basin in North Dakota, Minneso

South Dakota: U.S. Geological Survey Water-Reso

Investigations Report 02-4095, 271 p.

101.15s

2

qi

i = 1

n

∑

qi

i = 1

n

∑
Science Publishers, 522 p.

Macek-Rowland, K.M. and Dressler, V.M., 2002, Statistical 

summaries of water-quality data for selected streamflowgaging stations in the Red River of the North Basin, North 

Dakota, Minnesota, and South Dakota: U.S. Geological 

Survey Open-File Report 02-390, 229 p.

Miller, J.E. and Frink, D.L., 1984, Changes in flood response of 

the Red River of the North Basin, North Dakota-Minnesota: 

U.S. Geological Survey Water-Supply Paper 2243, 103 p.

Souris-Red-Rainy River Basins Commission, 1972, SourisRed-Rainy River basins comprehensive study: v. 3, 

Appendix D, p. D1-D157.

Tornes, L.H., Brigham, M.E., and Lorenz, D.L., 1997, 

Nutrients, suspended sediment, and pesticides in streams in 

the Red River of the North Basin, Minnesota, North Dakota, 

and South Dakota, 1993-95: U.S. Geological Survey WaterResources Investigations Report 97-4053, 70 p.

U.S. Geological Survey, 2000, Water resources data for North 

Dakota, water year 1999: U.S Geological Survey WaterData Report ND-99-1, 517 p.
Regression Equa

Selected Water-Gaging Stations North Dakota, M